3.1.79 \(\int e^{-x^2} x^3 \, dx\) [79]

Optimal. Leaf size=26 \[ -\frac {e^{-x^2}}{2}-\frac {1}{2} e^{-x^2} x^2 \]

[Out]

-1/2/exp(x^2)-1/2*x^2/exp(x^2)

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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2243, 2240} \begin {gather*} -\frac {1}{2} e^{-x^2} x^2-\frac {e^{-x^2}}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/E^x^2,x]

[Out]

-1/2*1/E^x^2 - x^2/(2*E^x^2)

Rule 2240

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^n*(
F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2243

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m
- n + 1)*(F^(a + b*(c + d*x)^n)/(b*d*n*Log[F])), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*((m + 1)/n)] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rubi steps

\begin {align*} \int e^{-x^2} x^3 \, dx &=-\frac {1}{2} e^{-x^2} x^2+\int e^{-x^2} x \, dx\\ &=-\frac {e^{-x^2}}{2}-\frac {1}{2} e^{-x^2} x^2\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 16, normalized size = 0.62 \begin {gather*} -\frac {1}{2} e^{-x^2} \left (1+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/E^x^2,x]

[Out]

-1/2*(1 + x^2)/E^x^2

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Maple [A]
time = 0.03, size = 21, normalized size = 0.81

method	result	size
gosper	\(-\frac {\left (x^{2}+1\right ) {\mathrm e}^{-x^{2}}}{2}\)	\(14\)
norman	\(\left (-\frac {x^{2}}{2}-\frac {1}{2}\right ) {\mathrm e}^{-x^{2}}\)	\(15\)
risch	\(\left (-\frac {x^{2}}{2}-\frac {1}{2}\right ) {\mathrm e}^{-x^{2}}\)	\(15\)
meijerg	\(\frac {1}{2}-\frac {\left (2 x^{2}+2\right ) {\mathrm e}^{-x^{2}}}{4}\)	\(18\)
derivativedivides	\(-\frac {{\mathrm e}^{-x^{2}}}{2}-\frac {x^{2} {\mathrm e}^{-x^{2}}}{2}\)	\(21\)
default	\(-\frac {{\mathrm e}^{-x^{2}}}{2}-\frac {x^{2} {\mathrm e}^{-x^{2}}}{2}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/exp(x^2),x,method=_RETURNVERBOSE)

[Out]

-1/2/exp(x^2)-1/2*x^2/exp(x^2)

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Maxima [A]
time = 2.14, size = 13, normalized size = 0.50 \begin {gather*} -\frac {1}{2} \, {\left (x^{2} + 1\right )} e^{\left (-x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/exp(x^2),x, algorithm="maxima")

[Out]

-1/2*(x^2 + 1)*e^(-x^2)

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Fricas [A]
time = 0.71, size = 13, normalized size = 0.50 \begin {gather*} -\frac {1}{2} \, {\left (x^{2} + 1\right )} e^{\left (-x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/exp(x^2),x, algorithm="fricas")

[Out]

-1/2*(x^2 + 1)*e^(-x^2)

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Sympy [A]
time = 0.02, size = 12, normalized size = 0.46 \begin {gather*} \frac {\left (- x^{2} - 1\right ) e^{- x^{2}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/exp(x**2),x)

[Out]

(-x**2 - 1)*exp(-x**2)/2

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Giac [A]
time = 0.64, size = 13, normalized size = 0.50 \begin {gather*} -\frac {1}{2} \, {\left (x^{2} + 1\right )} e^{\left (-x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/exp(x^2),x, algorithm="giac")

[Out]

-1/2*(x^2 + 1)*e^(-x^2)

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Mupad [B]
time = 0.11, size = 13, normalized size = 0.50 \begin {gather*} -\frac {{\mathrm {e}}^{-x^2}\,\left (x^2+1\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*exp(-x^2),x)

[Out]

-(exp(-x^2)*(x^2 + 1))/2

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